ENA, Section 1.4, Exercise 4b, n=2

Percentage Accurate: 75.1% → 100.0%

Time: 6.0s

Alternatives: 5

Speedup: 23.0×

• Unsound rule application detected in e-graph. Results may not be sound.

Specification

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{2} - {x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 2.0) (pow x 2.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 2.0) - pow(x, 2.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 2.0d0) - (x ** 2.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 2.0) - Math.pow(x, 2.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 2.0) - math.pow(x, 2.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 2.0) - (x ^ 2.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{2} - {x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 2.0) (pow x 2.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 2.0) - pow(x, 2.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 2.0d0) - (x ** 2.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 2.0) - Math.pow(x, 2.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 2.0) - math.pow(x, 2.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 2.0) - (x ^ 2.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x + x, \varepsilon, \varepsilon \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma (+ x x) eps (* eps eps)))

C

double code(double x, double eps) {
	return fma((x + x), eps, (eps * eps));
}

Julia

function code(x, eps)
	return fma(Float64(x + x), eps, Float64(eps * eps))
end

Wolfram

code[x_, eps_] := N[(N[(x + x), $MachinePrecision] * eps + N[(eps * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.6%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Step-by-step derivation

   1. unpow2 73.6%

      \[\leadsto \color{blue}{\left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right)} - {x}^{2} \]

   2. unpow2 73.6%

      \[\leadsto \left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right) - \color{blue}{x \cdot x} \]

   3. difference-of-squares 73.6%

      \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) + x\right) \cdot \left(\left(x + \varepsilon\right) - x\right)} \]

   4. *-commutative 73.6%

      \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right)} \]

   5. +-commutative 73.6%

      \[\leadsto \left(\color{blue}{\left(\varepsilon + x\right)} - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   6. associate--l+ 100.0%

      \[\leadsto \color{blue}{\left(\varepsilon + \left(x - x\right)\right)} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   7. +-inverses 100.0%

      \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   8. +-rgt-identity 100.0%

      \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   9. +-commutative 100.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

   10. associate-+r+ 100.0%

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

   11. count-2 100.0%

       \[\leadsto \varepsilon \cdot \left(\color{blue}{2 \cdot x} + \varepsilon\right) \]

   12. fma-def 100.0%

       \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]
4. Step-by-step derivation

   1. add-cbrt-cube 59.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)}} \]

   2. pow3 59.6%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)}^{3}}} \]

5. Applied egg-rr 59.6%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)}^{3}}} \]
6. Step-by-step derivation

   1. rem-cbrt-cube 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]

   2. fma-udef 100.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)} \]

   3. distribute-rgt-out 100.0%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \varepsilon + \varepsilon \cdot \varepsilon} \]

   4. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot x, \varepsilon, \varepsilon \cdot \varepsilon\right)} \]

   5. add-log-exp 73.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(e^{2 \cdot x}\right)}, \varepsilon, \varepsilon \cdot \varepsilon\right) \]

   6. *-commutative 73.1%

      \[\leadsto \mathsf{fma}\left(\log \left(e^{\color{blue}{x \cdot 2}}\right), \varepsilon, \varepsilon \cdot \varepsilon\right) \]

   7. exp-lft-sqr 73.1%

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(e^{x} \cdot e^{x}\right)}, \varepsilon, \varepsilon \cdot \varepsilon\right) \]

   8. log-prod 73.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(e^{x}\right) + \log \left(e^{x}\right)}, \varepsilon, \varepsilon \cdot \varepsilon\right) \]

   9. add-log-exp 75.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{x} + \log \left(e^{x}\right), \varepsilon, \varepsilon \cdot \varepsilon\right) \]

   10. add-log-exp 100.0%

       \[\leadsto \mathsf{fma}\left(x + \color{blue}{x}, \varepsilon, \varepsilon \cdot \varepsilon\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x + x, \varepsilon, \varepsilon \cdot \varepsilon\right)} \]

8. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x + x, \varepsilon, \varepsilon \cdot \varepsilon\right) \]

Alternative 2: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (fma 2.0 x eps)))

C

double code(double x, double eps) {
	return eps * fma(2.0, x, eps);
}

Julia

function code(x, eps)
	return Float64(eps * fma(2.0, x, eps))
end

Wolfram

code[x_, eps_] := N[(eps * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.6%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Step-by-step derivation

   1. unpow2 73.6%

      \[\leadsto \color{blue}{\left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right)} - {x}^{2} \]

   2. unpow2 73.6%

      \[\leadsto \left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right) - \color{blue}{x \cdot x} \]

   3. difference-of-squares 73.6%

      \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) + x\right) \cdot \left(\left(x + \varepsilon\right) - x\right)} \]

   4. *-commutative 73.6%

      \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right)} \]

   5. +-commutative 73.6%

      \[\leadsto \left(\color{blue}{\left(\varepsilon + x\right)} - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   6. associate--l+ 100.0%

      \[\leadsto \color{blue}{\left(\varepsilon + \left(x - x\right)\right)} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   7. +-inverses 100.0%

      \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   8. +-rgt-identity 100.0%

      \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   9. +-commutative 100.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

   10. associate-+r+ 100.0%

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

   11. count-2 100.0%

       \[\leadsto \varepsilon \cdot \left(\color{blue}{2 \cdot x} + \varepsilon\right) \]

   12. fma-def 100.0%

       \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]

4. Final simplification 100.0%

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right) \]

Alternative 3: 91.1% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-110} \lor \neg \left(x \leq 2.7 \cdot 10^{-100}\right):\\ \;\;\;\;\varepsilon \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -4.3e-110) (not (<= x 2.7e-100)))
   (* eps (* x 2.0))
   (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -4.3e-110) || !(x <= 2.7e-100)) {
		tmp = eps * (x * 2.0);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-4.3d-110)) .or. (.not. (x <= 2.7d-100))) then
        tmp = eps * (x * 2.0d0)
    else
        tmp = eps * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -4.3e-110) || !(x <= 2.7e-100)) {
		tmp = eps * (x * 2.0);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -4.3e-110) or not (x <= 2.7e-100):
		tmp = eps * (x * 2.0)
	else:
		tmp = eps * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -4.3e-110) || !(x <= 2.7e-100))
		tmp = Float64(eps * Float64(x * 2.0));
	else
		tmp = Float64(eps * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -4.3e-110) || ~((x <= 2.7e-100)))
		tmp = eps * (x * 2.0);
	else
		tmp = eps * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -4.3e-110], N[Not[LessEqual[x, 2.7e-100]], $MachinePrecision]], N[(eps * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.30000000000000025e-110 or 2.70000000000000016e-100 < x

   1. Initial program 38.6%

      \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   2. Step-by-step derivation

      1. unpow2 38.6%

         \[\leadsto \color{blue}{\left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right)} - {x}^{2} \]

      2. unpow2 38.6%

         \[\leadsto \left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right) - \color{blue}{x \cdot x} \]

      3. difference-of-squares 38.8%

         \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) + x\right) \cdot \left(\left(x + \varepsilon\right) - x\right)} \]

      4. *-commutative 38.8%

         \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right)} \]

      5. +-commutative 38.8%

         \[\leadsto \left(\color{blue}{\left(\varepsilon + x\right)} - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      6. associate--l+ 100.0%

         \[\leadsto \color{blue}{\left(\varepsilon + \left(x - x\right)\right)} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      7. +-inverses 100.0%

         \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      8. +-rgt-identity 100.0%

         \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      9. +-commutative 100.0%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

      10. associate-+r+ 99.9%

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

      11. count-2 99.9%

          \[\leadsto \varepsilon \cdot \left(\color{blue}{2 \cdot x} + \varepsilon\right) \]

      12. fma-def 99.9%

          \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]

   4. Taylor expanded in eps around 0 85.0%

      \[\leadsto \color{blue}{2 \cdot \left(\varepsilon \cdot x\right)} \]
   5. Step-by-step derivation

      1. *-commutative 85.0%

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} \]

      2. associate-*l* 85.1%

         \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \varepsilon} \]

      3. *-commutative 85.1%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(2 \cdot x\right)} \]

   6. Simplified 85.1%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(2 \cdot x\right)} \]

   if -4.30000000000000025e-110 < x < 2.70000000000000016e-100

   1. Initial program 97.1%

      \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   2. Step-by-step derivation

      1. unpow2 97.1%

         \[\leadsto \color{blue}{\left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right)} - {x}^{2} \]

      2. unpow2 97.1%

         \[\leadsto \left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right) - \color{blue}{x \cdot x} \]

      3. difference-of-squares 97.1%

         \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) + x\right) \cdot \left(\left(x + \varepsilon\right) - x\right)} \]

      4. *-commutative 97.1%

         \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right)} \]

      5. +-commutative 97.1%

         \[\leadsto \left(\color{blue}{\left(\varepsilon + x\right)} - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      6. associate--l+ 100.0%

         \[\leadsto \color{blue}{\left(\varepsilon + \left(x - x\right)\right)} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      7. +-inverses 100.0%

         \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      8. +-rgt-identity 100.0%

         \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      9. +-commutative 100.0%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

      10. associate-+r+ 100.0%

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

      11. count-2 100.0%

          \[\leadsto \varepsilon \cdot \left(\color{blue}{2 \cdot x} + \varepsilon\right) \]

      12. fma-def 100.0%

          \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]

   4. Taylor expanded in eps around inf 96.5%

      \[\leadsto \color{blue}{{\varepsilon}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 96.5%

         \[\leadsto \color{blue}{\varepsilon \cdot \varepsilon} \]

   6. Simplified 96.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-110} \lor \neg \left(x \leq 2.7 \cdot 10^{-100}\right):\\ \;\;\;\;\varepsilon \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 4: 100.0% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \varepsilon + \varepsilon \cdot \left(x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ (* eps eps) (* eps (* x 2.0))))

C

double code(double x, double eps) {
	return (eps * eps) + (eps * (x * 2.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * eps) + (eps * (x * 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return (eps * eps) + (eps * (x * 2.0));
}

Python

def code(x, eps):
	return (eps * eps) + (eps * (x * 2.0))

Julia

function code(x, eps)
	return Float64(Float64(eps * eps) + Float64(eps * Float64(x * 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * eps) + (eps * (x * 2.0));
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] + N[(eps * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.6%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Step-by-step derivation

   1. unpow2 73.6%

      \[\leadsto \color{blue}{\left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right)} - {x}^{2} \]

   2. unpow2 73.6%

      \[\leadsto \left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right) - \color{blue}{x \cdot x} \]

   3. difference-of-squares 73.6%

      \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) + x\right) \cdot \left(\left(x + \varepsilon\right) - x\right)} \]

   4. *-commutative 73.6%

      \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right)} \]

   5. +-commutative 73.6%

      \[\leadsto \left(\color{blue}{\left(\varepsilon + x\right)} - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   6. associate--l+ 100.0%

      \[\leadsto \color{blue}{\left(\varepsilon + \left(x - x\right)\right)} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   7. +-inverses 100.0%

      \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   8. +-rgt-identity 100.0%

      \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   9. +-commutative 100.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

   10. associate-+r+ 100.0%

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

   11. count-2 100.0%

       \[\leadsto \varepsilon \cdot \left(\color{blue}{2 \cdot x} + \varepsilon\right) \]

   12. fma-def 100.0%

       \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]
4. Step-by-step derivation

   1. fma-udef 100.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)} \]

   2. distribute-rgt-in 100.0%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \varepsilon + \varepsilon \cdot \varepsilon} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \varepsilon + \varepsilon \cdot \varepsilon} \]

6. Final simplification 100.0%

   \[\leadsto \varepsilon \cdot \varepsilon + \varepsilon \cdot \left(x \cdot 2\right) \]

Alternative 5: 72.9% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps eps))

C

double code(double x, double eps) {
	return eps * eps;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * eps
end function

Java

public static double code(double x, double eps) {
	return eps * eps;
}

Python

def code(x, eps):
	return eps * eps

Julia

function code(x, eps)
	return Float64(eps * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * eps;
end

Wolfram

code[x_, eps_] := N[(eps * eps), $MachinePrecision]

Derivation

1. Initial program 73.6%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Step-by-step derivation

   1. unpow2 73.6%

      \[\leadsto \color{blue}{\left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right)} - {x}^{2} \]

   2. unpow2 73.6%

      \[\leadsto \left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right) - \color{blue}{x \cdot x} \]

   3. difference-of-squares 73.6%

      \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) + x\right) \cdot \left(\left(x + \varepsilon\right) - x\right)} \]

   4. *-commutative 73.6%

      \[\leadsto \color{blue}{\left(\left(x + \varepsilon\right) - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right)} \]

   5. +-commutative 73.6%

      \[\leadsto \left(\color{blue}{\left(\varepsilon + x\right)} - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   6. associate--l+ 100.0%

      \[\leadsto \color{blue}{\left(\varepsilon + \left(x - x\right)\right)} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   7. +-inverses 100.0%

      \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   8. +-rgt-identity 100.0%

      \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   9. +-commutative 100.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

   10. associate-+r+ 100.0%

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

   11. count-2 100.0%

       \[\leadsto \varepsilon \cdot \left(\color{blue}{2 \cdot x} + \varepsilon\right) \]

   12. fma-def 100.0%

       \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]

4. Taylor expanded in eps around inf 70.5%

   \[\leadsto \color{blue}{{\varepsilon}^{2}} \]
5. Step-by-step derivation

   1. unpow2 70.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \varepsilon} \]

6. Simplified 70.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \varepsilon} \]

7. Final simplification 70.5%

   \[\leadsto \varepsilon \cdot \varepsilon \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=2"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 2.0) (pow x 2.0)))